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d) x >=4. (x+3) + (4-x) = (8+x) --> x = -1. We reject the solution because our condition is not satisfied (-1 is not more than 4)

thanks !

Responding to pm.

Absolute value properties:

When \(x\leq{0}\) then \(|x|=-x\), or more generally when \(some \ expression\leq{0}\) then \(|some \ expression|={-(some \ expression)}\). For example: \(|-5|=5=-(-5)\);

When \(x\geq{0}\) then \(|x|=x\), or more generally when \(some \ expression\geq{0}\) then \(|some \ expression|={some \ expression}\). For example: \(|5|=5\).

STEP BY STEP SOLUTION:

We have three transition points for \(|x+3| - |4-x| = |8+x|\): -8, -3, and 4 (transition point is the value of x for which an expression in the modulus equals to zero). Thus we have four ranges to check:

1. \(x<-8\);2. \(-8\leq{x}\leq{-3}\);3. \(-3<x<4\) 4. \(x\geq{4}\)

Note that it does not matter in which range(s) you include the transition points with "=" sign as long you include them.

1. When \(x<-8\), then \(x+3\) is negative, \(4-x\) is positive and \(8+x\) is negative. Thus \(|x+3|=-(x+3)\), \(|4-x|=4-x\) and \(|8+x|=-(8+x)\).

Therefore for this range \(|x+3| - |4-x| = |8+x|\): transforms to \(-(x+3) - (4-x) =-(8+x)\): --> \(x=-1\). This solution is NOT OK, since \(x=-1\) is NOT in the range we consider (\(x<-8\)).

2. When \(-8\leq{x}\leq{-3}\), then \(x+3\) is negative, \(4-x\) is positive and \(8+x\) is positive. Thus \(|x+3|=-(x+3)\), \(|4-x|=4-x\) and \(|8+x|=8+x\).

Therefore for this range \(|x+3| - |4-x| = |8+x|\): transforms to \(-(x+3) - (4-x) =8+x\): --> \(x=-15\). This solution is NOT OK, since \(x=-15\) is NOT in the range we consider (\(-8\leq{x}\leq{-3}\)).

3. When \(-3<x<4\), then \(x+3\) is positive, \(4-x\) is positive and \(8+x\) is positive. Thus \(|x+3|=x+3\), \(|4-x|=4-x\) and \(|8+x|=8+x\).

Therefore for this range \(|x+3| - |4-x| = |8+x|\): transforms to \(x+3 - (4-x) =8+x\): --> \(x=9\). This solution is NOT OK, since \(x=9\) is NOT in the range we consider (\(-3<x<4\)).

4. When \(x\geq{4}\), then \(x+3\) is positive, \(4-x\) is negative and \(8+x\) is positive. Thus \(|x+3|=x+3\), \(|4-x|=-(4-x)=x-4\) and \(|8+x|=8+x\).

Therefore for this range \(|x+3| - |4-x| = |8+x|\): transforms to \(x+3 - (x-4) =8+x\): --> \(x=-1\). This solution is NOT OK, since \(x=-1\) is NOT in the range we consider (\(x\geq{4}\)).

Where do the x = ' ' values come from? I have been staring at this for half an hour. I understood the whole concept in the '3-steps approach' but the '3-steps approach for complex problems' has me stuck suddenly. There goes mij GMAT-Mojo! Anyone able to help me get it back? Thanks.

BTW: Is this 650+ level?

You solve the equation to get the x = values

First of all, you are given |x+3|-|4-x|=|8+x|Convert this to |x+3|-|x-4|=|x+8| (since it is a mod, |4-x| is the same as |x-4|)

d) x >=4. (x+3) + (4-x) = (8+x) --> x = -1. We reject the solution because our condition is not satisfied (-1 is not more than 4)

thanks !

Responding to pm.

Absolute value properties:

When \(x\leq{0}\) then \(|x|=-x\), or more generally when \(some \ expression\leq{0}\) then \(|some \ expression|={-(some \ expression)}\). For example: \(|-5|=5=-(-5)\);

When \(x\geq{0}\) then \(|x|=x\), or more generally when \(some \ expression\geq{0}\) then \(|some \ expression|={some \ expression}\). For example: \(|5|=5\).

STEP BY STEP SOLUTION:

We have three transition points for \(|x+3| - |4-x| = |8+x|\): -8, -3, and 4 (transition point is the value of x for which an expression in the modulus equals to zero). Thus we have four ranges to check:

1. \(x<-8\);2. \(-8\leq{x}\leq{-3}\);3. \(-3<x<4\) 4. \(x\geq{4}\)

Note that it does not matter in which range(s) you include the transition points with "=" sign as long you include them.

1. When \(x<-8\), then \(x+3\) is negative, \(4-x\) is positive and \(8+x\) is negative. Thus \(|x+3|=-(x+3)\), \(|4-x|=4-x\) and \(|8+x|=-(8+x)\).

Therefore for this range \(|x+3| - |4-x| = |8+x|\): transforms to \(-(x+3) - (4-x) =-(8+x)\): --> \(x=-1\). This solution is NOT OK, since \(x=-1\) is NOT in the range we consider (\(x<-8\)).

2. When \(-8\leq{x}\leq{-3}\), then \(x+3\) is negative, \(4-x\) is positive and \(8+x\) is positive. Thus \(|x+3|=-(x+3)\), \(|4-x|=4-x\) and \(|8+x|=8+x\).

Therefore for this range \(|x+3| - |4-x| = |8+x|\): transforms to \(-(x+3) - (4-x) =8+x\): --> \(x=-15\). This solution is NOT OK, since \(x=-15\) is NOT in the range we consider (\(-8\leq{x}\leq{-3}\)).

3. When \(-3<x<4\), then \(x+3\) is positive, \(4-x\) is positive and \(8+x\) is positive. Thus \(|x+3|=x+3\), \(|4-x|=4-x\) and \(|8+x|=8+x\).

Therefore for this range \(|x+3| - |4-x| = |8+x|\): transforms to \(x+3 - (4-x) =8+x\): --> \(x=9\). This solution is NOT OK, since \(x=9\) is NOT in the range we consider (\(-3<x<4\)).

4. When \(x\geq{4}\), then \(x+3\) is positive, \(4-x\) is negative and \(8+x\) is positive. Thus \(|x+3|=x+3\), \(|4-x|=-(4-x)=x-4\) and \(|8+x|=8+x\).

Therefore for this range \(|x+3| - |4-x| = |8+x|\): transforms to \(x+3 - (x-4) =8+x\): --> \(x=-1\). This solution is NOT OK, since \(x=-1\) is NOT in the range we consider (\(x\geq{4}\)).

Thus no value of x satisfies \(|x+3| - |4-x| = |8+x|\).

Answer: A.

Hope it's clear.

Hi Bunuel,

I am getting confused at the sign changes. Please explain a bit clearly.

We have three transition points for \(|x+3| - |4-x| = |8+x|\): -8, -3, and 4 (transition point is the value of x for which an expression in the modulus equals to zero). Thus we have four ranges to check:

and confused for the other three ranges as well on how the expressions got +ve and -ve signs.What all need to be checked to give positive and negative signs to the expressions, as you mentioned in the solution for the above four ranges.

d) x >=4. (x+3) + (4-x) = (8+x) --> x = -1. We reject the solution because our condition is not satisfied (-1 is not more than 4)

thanks !

Responding to pm.

Absolute value properties:

When \(x\leq{0}\) then \(|x|=-x\), or more generally when \(some \ expression\leq{0}\) then \(|some \ expression|={-(some \ expression)}\). For example: \(|-5|=5=-(-5)\);

When \(x\geq{0}\) then \(|x|=x\), or more generally when \(some \ expression\geq{0}\) then \(|some \ expression|={some \ expression}\). For example: \(|5|=5\).

STEP BY STEP SOLUTION:

We have three transition points for \(|x+3| - |4-x| = |8+x|\): -8, -3, and 4 (transition point is the value of x for which an expression in the modulus equals to zero). Thus we have four ranges to check:

1. \(x<-8\);2. \(-8\leq{x}\leq{-3}\);3. \(-3<x<4\) 4. \(x\geq{4}\)

Note that it does not matter in which range(s) you include the transition points with "=" sign as long you include them.

1. When \(x<-8\), then \(x+3\) is negative, \(4-x\) is positive and \(8+x\) is negative. Thus \(|x+3|=-(x+3)\), \(|4-x|=4-x\) and \(|8+x|=-(8+x)\).

Therefore for this range \(|x+3| - |4-x| = |8+x|\): transforms to \(-(x+3) - (4-x) =-(8+x)\): --> \(x=-1\). This solution is NOT OK, since \(x=-1\) is NOT in the range we consider (\(x<-8\)).

2. When \(-8\leq{x}\leq{-3}\), then \(x+3\) is negative, \(4-x\) is positive and \(8+x\) is positive. Thus \(|x+3|=-(x+3)\), \(|4-x|=4-x\) and \(|8+x|=8+x\).

Therefore for this range \(|x+3| - |4-x| = |8+x|\): transforms to \(-(x+3) - (4-x) =8+x\): --> \(x=-15\). This solution is NOT OK, since \(x=-15\) is NOT in the range we consider (\(-8\leq{x}\leq{-3}\)).

3. When \(-3<x<4\), then \(x+3\) is positive, \(4-x\) is positive and \(8+x\) is positive. Thus \(|x+3|=x+3\), \(|4-x|=4-x\) and \(|8+x|=8+x\).

Therefore for this range \(|x+3| - |4-x| = |8+x|\): transforms to \(x+3 - (4-x) =8+x\): --> \(x=9\). This solution is NOT OK, since \(x=9\) is NOT in the range we consider (\(-3<x<4\)).

4. When \(x\geq{4}\), then \(x+3\) is positive, \(4-x\) is negative and \(8+x\) is positive. Thus \(|x+3|=x+3\), \(|4-x|=-(4-x)=x-4\) and \(|8+x|=8+x\).

Therefore for this range \(|x+3| - |4-x| = |8+x|\): transforms to \(x+3 - (x-4) =8+x\): --> \(x=-1\). This solution is NOT OK, since \(x=-1\) is NOT in the range we consider (\(x\geq{4}\)).

Thus no value of x satisfies \(|x+3| - |4-x| = |8+x|\).

Answer: A.

Hope it's clear.

Hi Bunuel,

I am getting confused at the sign changes. Please explain a bit clearly.

We have three transition points for \(|x+3| - |4-x| = |8+x|\): -8, -3, and 4 (transition point is the value of x for which an expression in the modulus equals to zero). Thus we have four ranges to check:

and confused for the other three ranges as well on how the expressions got +ve and -ve signs.What all need to be checked to give positive and negative signs to the expressions, as you mentioned in the solution for the above four ranges.

d) x >=4. (x+3) + (4-x) = (8+x) --> x = -1. We reject the solution because our condition is not satisfied (-1 is not more than 4)

thanks !

|x| = x when x >= 0 (x is either positive or 0)|x| = -x when x < 0 (note here that you can put the equal to sign here as well x <= 0 because if x = 0, |0| = 0 = -0 (all are the same) So the '=' sign can be put with x > 0 or with x < 0. We usually put it with 'x > 0' for consistency.

When we are considering ranges, say, x < -8 ------ x is less than -8-8 <= x < -3 ------- x is greater than or equal to -8 but less than -3-3 <= x < 4 ------- x is greater than or equal to -3 but less than 4x >=4 -------- x is greater than or equal to 4

We need to include the transition points (-8, -3, 4) somewhere so we include them with greater than sign.

In different sections of this number line, the terms are going to behave differently.When x < -8, |x + 8| = - (x + 8)For the other terms too, when we remove the absolute value sign, we need a negative sign.

At x = -8, the sign for |x +8| turns.When x >= -8 but less than -3, then |x+8| = x + 8 For the other terms, when we remove the absolute value sign, we need a negative sign.

Hence each of the four sections of the number line are considered separately.

What do you need to consider here? For |x - 2|, you have to think about x < 2 and x >= 2For |x - 5|, you have to think about x < 5 and x >= 5

To consider both together, you have one range x < 2. Another is x >= 2 and x < 5 so this is 2 <= x < 5. Yet another is x >= 5

Now, will you worry about x < 2 and at the same time, x > 5? No. There will not be such a value of x. This is where |x - 2| will open as -(x - 2) and |x - 5| will open as (x - 5). This doesn't exist.

So in a question like this: |x - 2| + |x - 5| = 4, you don't have 4 cases ((x - 2) positive or negative and (x - 5) positive or negative). You have only 3 cases. The 4th case will not exist.
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d) x >=4. (x+3) + (4-x) = (8+x) --> x = -1. We reject the solution because our condition is not satisfied (-1 is not more than 4)

thanks !

A faster way to solve this is to notice that:|x+3|-|4-x|=|8+x|>=0

this give us 2 equations:1. |x+3|-|4-x|>=0 - > solving 4 cases here (which 2 cases are not possible) give us x>=1/2 and x<=1/2 -> 1/2 is the only value for all the 4 options2. |8+x|>=0 -> give us x>=-8 or x<=-8 -> which give us 1 value x=1/2

Those 2 equations should have a shared area on the number line, and since they do no have it, there is no solution for this euqation -> 0 values.

Re: |x+3| - |4-x| = |8+x|. How many solutions does the equation
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29 Aug 2016, 05:45

johnwesley wrote:

You do it in the proper way. This method is called the "critical values" method. And once you have the critical values (by doing each absolute term equal to zero), you have to place them in the Real numbers line to make all the possible intervals. Then you just do the intervals as follows: x<lowest number in your real line, and then you take the intervals from each critical value to before the next one: i.e. x<-8, -8<=x<-3, -3<=x<4, x<=4. Therefore, you are getting all the possible intervals in the real line, and splitting the intervals from one critical value (including it) to before the next critical value (not including it).

Then, as you have done, you just set the predominant sign for each term under each condition, you solve the equation, and finally you check if the result saqtisfies the condition.

Re: |x+3| - |4-x| = |8+x|. How many solutions does the equation
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31 Aug 2016, 00:54

1

kshitij89 wrote:

johnwesley wrote:

You do it in the proper way. This method is called the "critical values" method. And once you have the critical values (by doing each absolute term equal to zero), you have to place them in the Real numbers line to make all the possible intervals. Then you just do the intervals as follows: x<lowest number in your real line, and then you take the intervals from each critical value to before the next one: i.e. x<-8, -8<=x<-3, -3<=x<4, x<=4. Therefore, you are getting all the possible intervals in the real line, and splitting the intervals from one critical value (including it) to before the next critical value (not including it).

Then, as you have done, you just set the predominant sign for each term under each condition, you solve the equation, and finally you check if the result saqtisfies the condition.

Now placing these values on the no. line we have four ranges & checking for each ranges :

1. For x greater than 4

Randomly selecting any value greater than 4 , lets say x = 10

|x+3| - |4-x| = |8+x|

|10+3| - |4-10| is not equal to |8+10|

So any value greater than 4 doesnot satisfy the equation.

2. For x between -3 & 4

Let x = 2 |x+3| - |4-x| = |8+x|

|2+3| - |4-2| is not equal to |8+2|

So any value between -3 & 4 doesnot satisfy the equation.

3. For x between -8 & -3,

Let x = -2

|x+3| - |4-x| = |8+x|

|-2+3| - |4-(-2)| is not equal to |8+(-2)|

So any value between -8 & -3 doesnot satisfy the equation.

4. For x lesser than -8,

Let x = -9

|x+3| - |4-x| = |8+x|

|-9+3| - |4-(-9)| is not equal to |8+(-9)|

So any value less than -8 doesnot satisfy the equation.

So the total solution is 0.Please confirm if my above approach is correct .

ThanksKshitij

Note that in this case, all values in the range will not satisfy the equation. Say when you take x < -8, you could have got x = -9. But what if you had actually tried x = -10 only and decided that no value in the range satisfies?
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Re: |x+3| - |4-x| = |8+x|. How many solutions does the equation
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16 Apr 2017, 18:46

1

this problem has a very simple solution, give me your kudos after reading it, thanks

so, first move the -[4-x] to the right,2nd step, square both sides of the equation3rd, take all variables to one side, simplify the result, (right now, one side should have 0)4th, you can see one side always is greater than 0, then the equation cannot hold, then there is no value of x

d) x >=4. (x+3) + (4-x) = (8+x) --> x = -1. We reject the solution because our condition is not satisfied (-1 is not more than 4)

thanks !

|x| = x when x >= 0 (x is either positive or 0)|x| = -x when x < 0 (note here that you can put the equal to sign here as well x <= 0 because if x = 0, |0| = 0 = -0 (all are the same) So the '=' sign can be put with x > 0 or with x < 0. We usually put it with 'x > 0' for consistency.

When we are considering ranges, say, x < -8 ------ x is less than -8-8 <= x < -3 ------- x is greater than or equal to -8 but less than -3-3 <= x < 4 ------- x is greater than or equal to -3 but less than 4x >=4 -------- x is greater than or equal to 4

We need to include the transition points (-8, -3, 4) somewhere so we include them with greater than sign.

in "a" when assumed that x is less than -8, the RHS modulus (8+x) is correctly negated. I am okay with this operation. Side by side, however, the LHS modulus (x + 3) is also negated...

what is the main logic underlying this negation ..? is this such that to make the LHS negative, some number should be subtracted from some bigger negatives..? if so, how can we conclude that (x+3) is bigger in value than is (4-x) ..?

d) x >=4. (x+3) + (4-x) = (8+x) --> x = -1. We reject the solution because our condition is not satisfied (-1 is not more than 4)

thanks !

|x| = x when x >= 0 (x is either positive or 0)|x| = -x when x < 0 (note here that you can put the equal to sign here as well x <= 0 because if x = 0, |0| = 0 = -0 (all are the same) So the '=' sign can be put with x > 0 or with x < 0. We usually put it with 'x > 0' for consistency.

When we are considering ranges, say, x < -8 ------ x is less than -8-8 <= x < -3 ------- x is greater than or equal to -8 but less than -3-3 <= x < 4 ------- x is greater than or equal to -3 but less than 4x >=4 -------- x is greater than or equal to 4

We need to include the transition points (-8, -3, 4) somewhere so we include them with greater than sign.

in "a" when assumed that x is less than -8, the RHS modulus (8+x) is correctly negated. I am okay with this operation. Side by side, however, the LHS modulus (x + 3) is also negated...

what is the main logic underlying this negation ..? is this such that to make the LHS negative, some number should be subtracted from some bigger negatives..? if so, how can we conclude that (x+3) is bigger in value than is (4-x) ..?

thanks in advance ...

wow..i think i have got it...

when x is less than -8, (x+3) got negative and hence this is negative ... hooray..

d) x >=4. (x+3) + (4-x) = (8+x) --> x = -1. We reject the solution because our condition is not satisfied (-1 is not more than 4)

thanks !

|x| = x when x >= 0 (x is either positive or 0)|x| = -x when x < 0 (note here that you can put the equal to sign here as well x <= 0 because if x = 0, |0| = 0 = -0 (all are the same) So the '=' sign can be put with x > 0 or with x < 0. We usually put it with 'x > 0' for consistency.

When we are considering ranges, say, x < -8 ------ x is less than -8-8 <= x < -3 ------- x is greater than or equal to -8 but less than -3-3 <= x < 4 ------- x is greater than or equal to -3 but less than 4x >=4 -------- x is greater than or equal to 4

We need to include the transition points (-8, -3, 4) somewhere so we include them with greater than sign.

in "a" when assumed that x is less than -8, the RHS modulus (8+x) is correctly negated. I am okay with this operation. Side by side, however, the LHS modulus (x + 3) is also negated...

what is the main logic underlying this negation ..? is this such that to make the LHS negative, some number should be subtracted from some bigger negatives..? if so, how can we conclude that (x+3) is bigger in value than is (4-x) ..?

thanks in advance ...

Consider this: If x < -8 (i.e. x can be -9, -10, -11, -11.4, -20 and so on...), what can you say about the sign of (x + 3) ?Can we say that when x is less than -3, (x + 3) is negative? Sure. So if we know that x is less than -8, then obviously, it is less than -3 too. So in this case, (x+3) will certainly be negative. Hence |x + 3| will also translate into -(x+3).

For (x + 3), all values of x to the left of -3 will give |x + 3| = -(x + 3). So if x < -8, it is to the left of -3 and hence will lead to |x+3| = -(x + 3).To the right of -3, all values of x will give |x + 3| = (x + 3).

Therefore for this range |x+3|−|4−x|=|8+x| |x+3|−|4−x|=|8+x|: transforms to x+3−(x−4)=8+x x+3−(x−4)=8+x: --> x=−1 x=−1. This solution is NOT OK, since x=−1 x=−1 is NOT in the range we consider (x≥4 x≥4).

Thus no value of x satisfies |x+3|−|4−x|=|8+x| |x+3|−|4−x|=|8+x|.

Hello Bunuel, Thanks for your reply.I can understand the solution for x>4 , which turns out to be x=-1 and not in the range we considered.But I am not able to understand the part when we take boundary value say x=4 , then 4-x=4-4=0 , and we know |x|=x ,x>=0, so why are we not considering this point in solution. So why are we considering |4-x| =-(4-x) only for x>4 ?I think I am missing something here .Please explain.

Re: |x+3| - |4-x| = |8+x|. How many solutions does the equation
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26 Dec 2017, 02:35

parthabar wrote:

Hello Bunuel, Thanks for your reply.I can understand the solution for x>4 , which turns out to be x=-1 and not in the range we considered.But I am not able to understand the part when we take boundary value say x=4 , then 4-x=4-4=0 , and we know |x|=x ,x>=0, so why are we not considering this point in solution. So why are we considering |4-x| =-(4-x) only for x>4 ?I think I am missing something here .Please explain.

The solution considers the range when x >= 4, so it must be true for x = 4 too bit let's still consider x = 4 case separately.

If x = 4, then |x+3|−|4−x|=|8+x| will be |4+3|−|4−4|=|8+4| --> |7|−|0|=|12| --> 7 = 12, which is NOT true, so x = 4 is not a solution for given equation. This is exactly what we got in the solution, when we considered x > = 4 range: NO solution in this range.
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